Fourth Proposition: The Plus of a Pawn does not always suffice to force the win, but in the majority of cases it does.
In considering this proposition, one must accentuate the condition of __ceteris paribus__ very strongly. The plan of "exchanging" the hostile pieces one by one, until the extra Pawn beside the two Kings remains alone on the board, is often difficult and perhaps impossible to be carried through. But, for all that, let us consider this task solved and let us now inquire into the concluding stages of that contest.
First, let the Pawn fight the adverse King unaided. Will the Pawn be able to advance unharmed on to the eighth row, there to be Queened, afterwards to Checkmate the King? Or will the King approach the Pawn meanwhile and capture it? The question is one of pure mathematics. While the Pawn advances one square the King approaches one sguare. Hence, the Pawn having to advance by Pawn steps until it Queens, the King to approach by King steps to the square where the Pawn Queens, all depends on the relation of the two members Pawn steps and King steps. If the number of Pawn steps is less than the number of King steps, the Pawn will Queen. If they are equal, or if the number of Pawn steps is greater than the number of King steps, the King will capture. For instance, if the White Pawn is on K6, White to move, the Black King must stand at a distance of two squares from White's K8, or the Pawn, though unaided, will Queen. The Black King must therefore in that moment stand upon one of the squares QB1, QB2, QB3, Q1, Q3, K1, K2, KB1, KB3, KX1, KX2, KX3 to stop the Pawn. The Black King must stand within a certain rectangle formed by two squares, which have the line White's K6, K7, K8 as a side. Each one of these squares is commonly spoken of as "the square of the Passed Pawn." The Pawn is "passed" because it has escaped the perils of opposing Pawns and is now free to advance to the eighth row unhampered by hostile pawns.
As an exercise show that if the Black King stands here on Q4, K4, or KB4, White having the move, it cannot stop the pawn.